HMMT 二月 2005 · 代数 · 第 4 题
HMMT February 2005 — Algebra — Problem 4
题目详情
- If a, b, c > 0, what is the smallest possible value of + + ? (Note that c a b b x c denotes the greatest integer less than or equal to x .)
解析
- If a, b, c > 0, what is the smallest possible value of + + ? (Note that c a b b x c denotes the greatest integer less than or equal to x .) Solution: 4 Since b x c > x − 1 for all x , we have that ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ a + b b + c c + a a + b b + c c + a
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- − 3 c a b c a b ( ) ( ) ( ) a b b c c a = + + + + + − 3 . b a c b a c But by the AM-GM inequality, each of the first three terms in the last line is at least
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